Line up the decimal commas — then add exactly like whole numbers
The Golden Rule
Always line up the decimal commas (not the digits). Add zeros to make all numbers the same length. Then add column by column, right to left, carrying when needed.
Place Value Reminder
Hundreds
Tens
Units
●
Tenths
Hundredths
Thousandths
100
10
1
,
\(\frac{1}{10}\)
\(\frac{1}{100}\)
\(\frac{1}{1000}\)
Example 1 — Simple addition
Calculate \(12{,}67 + 4{,}76\)
1
Line up commas: \(\phantom{0}12{,}67\) \(+\phantom{0}4{,}76\) \(\overline{\phantom{0000000}}\)
Calculate: \(8{,}965 + 4{,}236\) — type your answer using a comma
Line up: 8,965 + 4,236. Add column by column right to left.
➖Subtraction of Decimal Fractions
Same as addition — line up commas, fill zeros, subtract with borrowing
The Rule
Line up decimal commas. Fill in zeros so both numbers have the same number of decimal places. Subtract right to left, borrowing from the next column when needed.
Example 1 — Straightforward
Calculate \(18{,}08 - 3{,}87\)
1
Hundredths: \(8-7=1\)
2
Tenths: \(0-8\) → can't! Borrow from units: \(10-8=2\)
⚠️ Tip: Subtracting from a whole number like \(5 - 1{,}37\)? Write it as \(5{,}00 - 1{,}37\) first.
Check Your Understanding
Calculate: \(18{,}98 - 8{,}96\)
Calculate: \(8{,}965 - 4{,}236\)
Calculate: \(10 - 3{,}625\) — type your answer using a comma
Write as \(10{,}000 - 3{,}625\). Borrow as needed.
✖️Multiplication of Decimal Fractions
Drop the commas, multiply as whole numbers, then place the comma back
The Method — Count Decimal Places
Step 1: Count the total number of decimal places in both numbers combined. Step 2: Multiply ignoring the commas (treat as whole numbers). Step 3: Place the comma so the answer has the same total decimal places.
\(1{,}2 \times 0{,}35\)
1
Decimal places: 1 + 2 = 3
2
\(12 \times 35 = 420\)
3
3 d.p. → \(0{,}420 = 0{,}42\)
\(3{,}25 \times 0{,}75\)
1
Decimal places: 2 + 2 = 4
2
\(325 \times 75 = 24\,375\)
3
4 d.p. → \(2{,}4375\)
Multiplying by 10, 100, 1 000
Multiply by 10 → move comma 1 place right |
by 100 → 2 places |
by 1 000 → 3 places
\(0{,}3 \times 0{,}4 \times 100\): first multiply: \(0{,}3 \times 0{,}4 = 0{,}12\), then \(\times 100 = 12\)
Check Your Understanding
Calculate: \(5{,}9 \times 0{,}42\)
Calculate: \(0{,}15 \times 6{,}5\)
Calculate: \(0{,}3 \times 0{,}5 \times 10\) — type your answer
\(0{,}3 \times 0{,}5 = 0{,}15\), then \(\times 10 = 1{,}5\)
➗Division of Decimal Fractions
Divide by a whole number directly; divide by a decimal by converting first
Case 1 — Dividing by a Whole Number
Keep the comma in line with the dividend. Divide as with whole numbers.
\(0{,}8 \div 4\)
1
\(8 \div 4 = 2\), one decimal place
2
\(0{,}2\)
\(0{,}81 \div 9\)
1
\(81 \div 9 = 9\), two decimal places
2
\(0{,}09\)
Case 2 — Dividing by a Decimal
Multiply both numbers by a power of 10 to make the divisor a whole number. The answer does not change.
\(6{,}3 \div 0{,}21\)
1
Multiply both by 100: \(630 \div 21\)
2
\(630 \div 21 = \) 30
\(3{,}75 \div 0{,}5\)
1
Multiply both by 10: \(37{,}5 \div 5\)
2
\(37{,}5 \div 5 = \) 7,5
How many times does 0,25 go into 5,5?
1
\(5{,}5 \div 0{,}25\) — multiply both by 100: \(550 \div 25\)
2
\(550 \div 25 = \) 22
💡 Tip — which power of 10? Count the decimal places in the divisor (the number you're dividing BY). Use that many zeros: 1 d.p. → ×10, 2 d.p. → ×100, 3 d.p. → ×1000.
Check Your Understanding
Calculate: \(0{,}54 \div 6\)
Calculate: \(36 \div 0{,}2\)
Calculate: \(5{,}5 \div 0{,}25\) — type your answer
Multiply both by 100: \(550 \div 25\)
🔢Squares, Cubes & Roots of Decimals
Square and cube = repeated multiplication · Roots = convert to fraction first
Squaring a Decimal
\(a^2 = a \times a\). Count decimal places in \(a\), double them for \(a^2\).
\((0{,}7)^2\)
1
\(0{,}7 \times 0{,}7\)
2
\(7 \times 7 = 49\), total d.p. = 1+1 = 2
3
\(0{,}49\)
\((0{,}12)^2\)
1
\(0{,}12 \times 0{,}12\)
2
\(12 \times 12 = 144\), d.p. = 2+2 = 4
3
\(0{,}0144\)
Cubing a Decimal
\(a^3 = a \times a \times a\). Decimal places in \(a^3\) = decimal places in \(a\) × 3.
\((0{,}1)^3\)
1
\(0{,}1 \times 0{,}1 \times 0{,}1\)
2
\(1 \times 1 \times 1 = 1\), d.p. = 1+1+1 = 3
3
\(0{,}001\)
\((0{,}04)^3\)
1
\(4 \times 4 \times 4 = 64\), d.p. = 2+2+2 = 6
2
\(0{,}000064\)
Square Root of a Decimal
Convert to a common fraction first. Find the square root of numerator and denominator separately. Only works neatly with perfect squares.
💡 Pattern to remember:
\(\sqrt{\phantom{0}}\) of a decimal with 2 d.p. → answer has 1 d.p.
\(\sqrt{\phantom{0}}\) of a decimal with 4 d.p. → answer has 2 d.p.
\(\sqrt[3]{\phantom{0}}\) of a decimal with 3 d.p. → answer has 1 d.p.
\(\sqrt[3]{\phantom{0}}\) of a decimal with 6 d.p. → answer has 2 d.p.
Check Your Understanding
Calculate: \((0{,}4)^2\)
Calculate: \(\sqrt{0{,}25}\)
Calculate: \(\sqrt[3]{0{,}125}\) — type your answer using a comma
\(0{,}125 = \dfrac{125}{1000}\). Find \(\sqrt[3]{125}\) and \(\sqrt[3]{1000}\).